论文标题
由惩罚方法引起的线性系统的条件
Conditioning of linear systems arising from penalty methods
论文作者
论文摘要
在轻度假设下，惩罚不可压缩性问题导致条件编号$κ= \ MATHCAL {O}（\ VAREPSILON ^{ - 1} h ^{ - 2}）$，$ \ \ varepsilon = $ varepsilon = $ nocal poninty参数$ << 1 $，$ h = $ mesh width $ <1 $ <1 $ <1 $ <1 $ <1 $ <1 $ <1 $ <1 $ <1 $ <1 $ <尽管$κ= \ MATHCAL {O}（\ Varepsilon ^{ - 1} H ^{ - 2}）$是大型的，但实际的测试很少报告难以解决这些系统。在SPD情况下，使用共轭梯度方法，通常是通过惩罚系数矩阵中发生的光谱差异来解释的。在此，我们指出了第二个促成因素。由于该解决方案大致不可压缩，因此与罚款项相关的特征空间中的解决方案组件可能很小。结果，有效条件号可能比标准条件编号小得多。
Penalizing incompressibility in the Stokes problem leads, under mild assumptions, to matrices with condition numbers $κ=\mathcal{O} (\varepsilon ^{-1}h^{-2})$, $\varepsilon =$ penalty parameter $<<1$, and $ h= $ mesh width $<1$. Although $κ=\mathcal{O}(\varepsilon ^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solving these systems. In the SPD case, using the conjugate gradient method, this is usually explained by spectral gaps occurring in the penalized coefficient matrix. Herein we point out a second contributing factor. Since the solution is approximately incompressible, solution components in the eigenspaces associated with the penalty terms can be small. As a result, the effective condition number can be much smaller than the standard condition number.
